Edge-connected, crossed-electrode array for two-dimensional projection and beamforming

A ness geometr for array construction is discussed sshich TN TOP ELECTRODES employs tso sets of orthogonal striped electrodes. While obtaining A'T2 T3 intersecting points in a Iso-dimensional structure. onl% 2N\ control points are required. Thus. Vactive elements are controlled ,,ith 2A' at r r -1 F i fl degrees of freedom hich both simplifies implementation and data LT I L _l -TJI handling. This gain in simplicito is traded off against reduced perforB2 = = El mance -Ahen emploed as a projector and increased signal processing 113 -: I i LtA -Ahen emplo*ed in beamforming mode. It is shosn that no performance 8."It 1 ; ; r . is lost in beamforming if the process is carried out in a tEso-%tep pro' 1 Fn ces. I himitations of the crossed-electrode geometr% are discussed F7 n n L . I H .. and a theorY is presented for operation as a projector and a receiver. F_ F__ -IN RODUCTION HE purpose of this paper is the introduction of a new geL_J L:l 7I F-1 Fn r----F-I F -I F-71 Lometrx for planar structures which allo\s simpler imple__JL: ,_L =Jl_.__ mentation and construction while trading off higher signal I \ _ processing costs and possible reduced performance (the tern' BOTTOM ELECTRODES ACTIVE AREA CROSS-POINT planar is not strictly necessary although most applications are concerned with planar devices). Since to-dimensional strucaTOP ELECRODES tures are important in acoustic arra\s and optical imaging devices, the structure described here has applications which cover __I_________,_________________[___ a wide variet. of technical areas. In this analysis, we will give . I the basic operation principles of the geometry and leave det-i!-d analysis of specific applications to a forum where the I physics of each may be dealt with in detail. MATERIAL BOTTOM ELECTOOES The generic physical description of our device is shown in h, Fig. Ib). It is assumed that an active material (acoustical or Fig. Tvo-dimcnional croed-elcctrodc arra., (a) Top "',c.. 1h) utoptical) is placed bet-,een Ixo electrodes as shown. By applya\a\ ide \,ic%\ ing a ,oltage. the de\ ice is able to act as a projector of acoustical or optical cncrg,, or. left in a passive state. it ma\ create mentation. the active element is sandw, iched bet\cen electrodes a voltage by the action of a field (acoustic or optical) impinging and each max be addressed individually either bN a random adupon it. Obviously. this is a simplification to any real device, dress scheme where each pair of electrodes leads is a\ailable, but for the purposes of describing a new geometrical impleor through a parallel-to-serial conkersion such as in charge colmentation for a two-diniensional arra%, it is suflicient. The acpled devices (CCD). The problems of individual access of each tive material could be a piezoelectric material for acoustical element are mx riad: large numbers of connections. large numapplications (pol,,vinylidenc fluoride PVDF). a nematic liquid hers of wircs. greater potential for failure. etc. The heat loss crystal or light emiting diode material (GaAs) for optical proalong conducting tires in cooled detectors, and the inability to Jection applications or a Si-based optical imaging device. There package a random array were two maJor motivations for the exists a plethora of materials and applications which we will de'elopment of the parallel-to-serial conversion devices such as not discuss. howevcr, we do make the realistic assumption that CCI's. The basic problem is that for an N x N device..V' all electrodes are transparent to the type of field (acoustic or elements must be addressed. The contribution of this paper is optical) under consideration. to suggest another t pc of structure, that of crossed clectroded Fig. I showxs a two-dimensional structure which is typical of strips as shown in Fig. I. Here the device has one set of electhe planar arrays under consideration. In a conventional impletrode strips on top of the active material, and another orthogonal set on the bottom. This leaves N" intersection points while resulting in only 2N electrodes which are brought out to the Manu,,rtp rcccm,cd August 14. 19XX; rcx.i,,d JanTuIr) 21. 190!. ed.Ob'ul.anray, h2Negesffedo DF The author "kas . ith the Undcratcr Sound Reference Detachment. Naedge. Obvtously. an array \% ith 2N degrees of freedom DOF %al Rcscarch 1.ahorator%. Orlant o. Ff. 32856 9337 will not function as efficiently as one with N. but if one conIF-E Log Number 90411 3h siders the savings for a device \with. for example. I(' element, 1053.597X 91 0201-t128950)1 .1) 1991 Il:tI IEEE TRANSACTIONS ON SIGNAL PROCESSING, VOL. 39 NO. 2. FEBRUARY 1991 per side (2 x 10' versus 10") this loss of pertormance may be small number is less than the dimensionality of the autocorreacceptable. lation matrix. We will analyze this structure through reciprocity. i.e.. as a Whereas for two-dimensional field: without correlation, each projector first, later as a receiver for beamfoning. We will neeigenvalue provides the same amount of information for deglect crosstalk which can be present in a device with either 2N scribing the field in, terms of an expansion of eigenvectors. with or N DOF. Given the results shown in subsequent sections on finite correlation each eigenvalue provides proportionally more projection and beamforming, a decision can be made on the information than the next smaller one, so that truncation of the feasibility of employing a crossed-electrode edge-coupled array expansion after a few terms yields an accurate description 151. in specific acoustical or optical applications. Thus practically speaking, the required number of DOF to expand a function which has finite correlation properties. is less Two-DIMENSIONAL EDGE-CONTROLLE) ARRAY than the dimensionality of the problem if the expansion is allowed to have a finite but small mean-squared error. Since even Consider Fig. 1. The device pictured has an active material full expansions will have some variance from the original funcsandwiched between two sets of electrodes. The top set consists lion due to noise, this limited DOF expansion may not be an of a number N of strips, while the bottom has a similar pattern overwhelming constraint. rotated 900. The problem of approximating N 2 independent potential difBy applying a voltage to e:;ch top electrode and each bottom ferences by 2N edge voltages may be represented in matrix form electrode, a potential difference at the intersection of the ith as bottom andjth top electrodes of (B, T, ) is formed. Electric field strength may be approximated as this potential difference AV = W (2) divided by the electrode separation, and the emitted field will where V is the array of edge voltages, bottom voltage first. folbe assumed to be the product of the electric field and a constant lowed by top voltages. For N channels it takes the form (see which describes the material. In acoustics, this constant would Fig. I). be the piezoelectric e/ constant, whereas in optics it is the diB, electric c constant for liquid crystals or another conversion facB tor which relates light intensity to voltage for devices such as a B. biased light emitting diodes operated over their linear regions. The figure indicates the active area to be the spatial intersection of two rectangular electrodes. Actually, fringing of the fields By will cause a response pattern to fall off more gradually than the V = 2N elements. (3) perfectly rectangular shape shown. This structure is both physT, ically realizable and controllable within current signal processing technology. T, Questions which remain are what types of fields may be created by such a device, and the possibility of employing such a device as a receiver. Additional investigations must center LTA. around control stability, possible implementation problems. and--methodology for possible improvements. The array W is the array of two-dimensional potential differences ordered row-wise (the array W is proportional to the desired two-dimensional field on the surface of the array) TH EORY-PRoJECTION The field generated from each active area is linearly proportional to the electric field, which is nearly proportional to the M12 potential difference at that location. Then the potential difference at any intersection point where the ith bottom electrode crosses thejth top electrode may be written as M, W = MI N2 elements (4) M, = BT( M where B. Tare the top and bottom electrode voltages for the ith row and jth column. Notice that while the number of active areas of an N X N structure device has N (DOF). the edgeinterconnected structure reduces this to 2N DOF thus limiting and the matrix A has the form the types of patterns which may be represented. It is expected 2N eleiients that the desired two-dimensional spatial fields have finite cor11 relation lengths and times, which will reduce the number of I 0 0 0 . I 0 0 0 DOF required to describe their spatial and temporal behavior. I 0 0 0 . 0 -1 0 0 ... An alternate position from which to view this is to consider the description an isotropic field (isotropic assumption is not necA N: elements. essary but simplifies the discussion) in terms of its Karhunen0 I 0 0 I 0 0 0 Loeve (K-L) expansion 111-141. The diagonalization of the autocorrelation (expanding in terms of its cigenfunctions) matrix typically shows that the autocorrelation matrix is not of full rank 0 0 .. 1 0I 0 0 0 ••and the number of cigenvalues which exceed some arbitrarily (5) lyial hw httcatceainmti sni ffl ak00i 0 (1 1) ( SCHAU: EDGE-CONNECTED, CROSSED-ELECTRODE ARRAY 291 Since A has more rows than columns the generalized inverse is where I is the (N x N ) matrix consisting of all ones, 0 the (N AN = (A'A)I A' (overdetermined case) (6) x N) null matrix. The pseudoinverse is then written as I , I and it is well known that the solution to (2) takes the general B" (BrB)'BI = I I + I $ B I form (if the original equation is consistent) 171 N 2 V=AnW + (I A*A)Z (7) B 1 1 -+ Q(15) where Af is the Penrose-Moore inverse and Z is arbitrary. We choose a solution of the form -N -1 -1 -1 ' V= A"W (8) -N -1 -1 -1 which given the form for A in (6) for the overdetermined case, the solution (8) is best in a least squares sense. Since the quantities of interest are voltage differences between the rows and columns, it is convenient to introduce a normalization which fixes the absolute voltage. This can be Q -N -1 -I -1 (16) achieved in a variety of ways, one which is sufficiently general is to write N ,. -N I I I SB, + T, = C (9) (9) -N I I I i.e., fix the sum of all edge voltages at some arbitrary constant C. Other forms of normalization can be found from this approach. This adds a constraint equation to (2) and changes the form of A and W. Writing the augmented equation in the form The final solution may be written in terms of the nonaugmented operators as BV= T B A T + VOG H--G W+ K (17) BT [C] (10) N 2N-I A I where where A and Ware given by (5) and (4), respectively. Equation (10) has solution -1 -I -I -I V = B*T; B= (BTB)-BT. (11) In this formulation G= -1 -1 -1 -1 ... (18)